The drain plug in your bathroom sink operates through a linkage system under the sink. The pull lever behind the faucet moves the plug up and down using this linkage. The plug has a ring on the bottom that the end of the linkage inserts into to move the plug. When you need to remove the plug from your bathroom sink, disconnect the piece of the linkage that inserts in the bottom ring of the plug.
Slide under the sink and locate the linkage from the pull lever to the sink drain pipe. The horizontal part of the linkage goes into the sink drain pipe below the sink basket. The horizontal part connects to the pipe with a retaining nut.
Turn the retaining nut clockwise with a pair of slip-joint pliers. Turn the nut clockwise because when you are facing the sink, you will be facing the nut in reverse.
Pinch the clip that secures the horizontal linkage to the vertical linkage and pull the horizontal linkage toward the back wall to pull it out of the sink drain pipe.
Pull the plug out of the sink drain.